MT1710: WEEK 4 - 2018/19 Section 1.4: Curve Sketching (continuation). Example 1.18 (continuation) Sketch the following curves, using the given guidelines : (e) y = x2+2x+1 , (f) y = ; @2-x+1' x(x+1) (g) ay2= x(a2 - x2), a> 0, (h) a2y2 = x2(a2 - x2) , (i) y2= x2(x-a), a > 0, (j) ay2 = x3, a> 0. (e) Symmetry? Origin on the curve? Intersections with the axes? Behaviour near the origin? Asymptotes? Does curve intersects asymptotes? Gradient is easy enough here, so what does it confirm? (f) Symmetry? Origin on the curve? Intersections with the axes? Behaviour near the origin? Asymptotes? Does curve intersects asymptotes? Gradient is easy enough here, so what does it confirm? (g) Symmetry? Origin on the curve? Intersections with the axes? Excluded regions? Behaviour near the origin? Behaviour for large values of x?
(h) Symmetry? Origin on the curve? Intersections with the axes? Excluded regions? Behaviour near the origin? (i) Symmetry? Origin on the curve? Intersections with the axes? Excluded regions? Behaviour for x large? (j) Symmetry? Origin on the curve? Intersections with the axes? Excluded regions? Behaviour for x large? Chapter II: Differentiation Section 2.1: Introduction Definition 2.1 The function f is said to be differentiable at the point x or f is said to have a derivative at x whenever lim §x ->0 f(x+0x) - f(x) exists as a finite real number. Otherwise it is said to be non-differentiable at x. Recall that the first derivative of the function f at x is equal to the gradient of the tangent line to the graph of f at x. Basic Rules: I: Linearity df dx (af + bg) = a + b- dg ; dx d dx
II: Product Rule a (fg) = 29 + f Tr; dg III: Quotient Rule d dx g -(+) = g g2 df _ fdg , g ¥0; IV: Chain Rule If z = f(y) and y = g(x), then dx dz dy dx ' dz dy for a and b arbitrary real constants and differentiable functions f and g. Example 2.1 Let u and v be functions of x and let y = uv. Derive the product rule. Example 2.2 Evaluate the derivatives of x3 sin x and x2 sin x cos x. Example 2.3 Evaluate the derivatives of tan x and cosec x. Examples 2.4 Evaluate the derivatives of (i) (x4 + a4)6 , (ii) cos(x3), (iii) (cos x)3 , (iv) (2) 2 (a2 - x2) } . In each case, a may be treated as a constant.